3.2 BEC critical temperature

Chapter 3

Bose-Einstein Condensation of An Ideal Gas

An ideal gas consisting of non-interacting Bose particles is a fictitious system since every realistic Bose gas shows some level of particle-particle interaction. Nevertheless, such a mathematical model provides the simplest example for the realization of Bose-Einstein condensation. This simple model, first studied by A. Einstein [1], correctly describes important basic properties of actual non-ideal (interacting) Bose gas. In particular, such basic concepts as BEC critical temperature T_c (or critical particle density n_c), condensate fraction N₀/N and the dimensionality issue will be obtained.

3.1 The ideal Bose gas in the canonical and grand canonical ensemble

Suppose an ideal gas of non-interacting particles with fixed particle number N is trapped in a box with a volume V and at equilibrium temperature T . We assume a particle system somehow establishes an equilibrium temperature in spite of the absence of interaction.

Such a system can be characterized by the thermodynamic partition function of canonical ensemble

Z =X

R

e^−βER, (3.1)

where R stands for a macroscopic state of the gas and is uniquely specified by the occupa- tion number n_i of each single particle state i: {n₀, n₁, · · · ·}. β = 1/k_BT is a temperature parameter. Then, the total energy of a macroscopic state R is given by only the kinetic energy:

E_R=X

i

ε_in_i, (3.2)

where ε_i is the eigen-energy of the single particle state i and the occupation number n_i satisfies the normalization condition

N =X

i

n_i. (3.3)

The probability of finding the ideal gas in a specific macroscopic state R is given by the law of large numbers in statistical mechanics [2]:

P_R= e^−βER

Z . (3.4)

Using (3.4), one can calculate the average particle number of a specific single particle state (s)

n_s = X

R

n_sP_R

= P

{n0,n1···}n_se^{−β(ε0n0+ε1n1+···)} P

{n0,n1···}e^{−β(ε0n0+ε1n1+···)}

= P

nsn_se^−βεsnsP_(s)

e^{−β(ε0n0+ε1n1+···)} P

nse^−βεsnsP_(s)

e^{−β(ε0n0+ε1n1+···)} (3.5) HereP_(s)

stands for the summation over {n₀, n₁, · · ·} except for a particular summation for the state s. If the state s does not have a particle, N particles must be distributed over the states other than s so that the partition function in this case is given by

Z_s(N ) = X(s)

e^{−β(ε0n0+ε1n1+···)}, (3.6)

X(s) i

n_i= N. (3.7)

If the state s has one particle, the remaining N − 1 particles must be distributed over the states other than s and the partition function in this case is modified to

Z_s(N − 1) = X(s)

e^{−β(ε0n0+ε1n1+···)}, (3.8) X(s)

i

n_i = N − 1. (3.9)

If we substitute these expressions into (3.5), we have

n_s= 0 × Z_s(N ) + e^−βεsZ_s(N − 1) + 2e^−2βεsZ_s(N − 2) + · · ·

Z_s(N ) + e^−βεsZ_s(N − 1) + e^−2βεsZ_s(N − 2) + · · · . (3.10) In order to proceed further, one has to introduce the approximate expression for the partition functions Z_s(N − ∆N ). For this purpose, calculating the logarithmic function log Z_s(N − ∆N ) is more appropriate than calculating Z_s(N − ∆N ) itself, since the former is a much more slowly varying function of ∆N compared to the latter [2]. Then one can truncate the expansion of log Z_s(N − ∆N ) to the first order of ∆N :

log Z_s(N − ∆N ) ' log Z_s(N ) +

· ∂

∂N log Z_s(N )

¸

(−∆N ). (3.11)

The first-order expansion coefficient is considered to be a universal constant independent of s:

α_s= ∂

∂N log Z_s(N ) = ∂

∂N log Z(N ) = α. (3.12)

This is because Z(N ) is the summation over many microscopic states and so the variation of its logarithm with respect to the total number of particles should be insensitive as to weather or not the particular state s is omitted. From (3.11) and (3.12), one obtains

Z_s(N − ∆N ) ' Z_s(N )e^−α∆N. (3.13) Using (3.13) in (3.10), one can calculate the average occupation number of the state s as

n_s = P

nsn_se^{−ns(βεs+α)} P

nse^{−ns(βεs+α)}

= −1 β

∂

∂ε_slog

"

X

s

e^{−ns(βεs+α)}

#

= 1

e^βεs+α− 1. (3.14)

The first-order expansion coefficient α is now replaced by a new parameter, called the chemical potential defined by µ = −α/β.µ can be uniquely determined by the normaliza- tion condition (3.3):

N =X

i

n_i =X

i

1

e^β(εi−µ)− 1. (3.15)

Equation (3.14) is referred to as Bose-Einstein distribution function, in which the average occupation number n_sis determined uniquely by the temperature parameter β, the eigen- energy of the single particle state ε_s and the chemical potential µ.

The thermodynamic partition function (3.1) was defined for the system with a fixed number of particles. By taking an advantage of the unique relationship (3.15) between the total number of particles N and the chemical potential µ, one can extend (3.1) to the system with a varying total number of particles. The new thermodynamic partition function can be written as

Z =X

n0

e^{−β(ε0−µ)n0}X

n1

e^{−β(ε1−µ)n1}· · · . (3.16) By considering the chemical potential µ as a free parameter, (3.16) can describe the system with all possible values of N and is referred to as the thermodynamic partition function of grand canonical ensemble.

The grand canonical potential can be defined by Ω = E − T S − µN , where E and S are the total energy and the entropy of the system[2]. Then, Ω is related to Z via [2]

Ω = −k_BT lnZ = k_BTX

i

ln h

1 − e^{−β(εi−µ)} i

. (3.17)

The total number of particles N can be evaluated from (3.17) as N = −∂Ω

∂µ =X

i

1

e^β(εi−µ)− 1, (3.18)

as well as the total energy

E = Ω − T∂Ω

∂T − µ∂Ω

∂µ =X

i

ε_i

e^β(εi−µ)− 1. (3.19) as they should be. If we recall the average occupation number of the state i is n_i =

1

e^β(εi−µ)−1, the expressions (3.18) and (3.19) can be easily interpreted. The entropy S is similarly calculated as

S k_B ≡ 1

k_B µ

−∂Ω

∂T

¶

=X

i

[log (1 + n_i) + n_ilog (1 + n_i)] . (3.20) This is the maximum entropy (randomness) if the average number of particles per state n_i is given. The probability distribution of each state is that of a thermal state, i.e.

p_i(n) = _1+n¹ _i

³ ni

1+ni

´_n

[2]. The first term of R.H.S in (3.20) dominates the entropy when the single particle state is occupied by many particles, n_iÀ 1, and is referred to as ”wave entropy” [3]. The second term of R.H.S. in (3.20) dominates the entropy when the single particle state is occupied sparsely, n_i ¿ 1, and is referred to as ”particle entropy” [3].

3.2 BEC critical temperature

Equation (3.14) provides the important constraint for the chemical potential, i.e. µ < ε₀, where ε₀ is the lowest energy eigenvalue of the single particle states. The violation of this inequality would result in a negative value of the occupation number for the states with energies smaller than µ. When µ approaches ε₀ from smaller values, the occupation number n₀ = _{eβ(ε0−µ)}¹ ₋₁ becomes increasingly large. This is the physical picture behind Bose-Einstein condensation. One can split the total number of particles into

N = N₀+ N_th, (3.21)

where

N_th =X

i6=0

n_i(T, µ), (3.22)

is the number of particles in all excited states except for the lowest energy ground state.

N_this also referred to as the thermal component of the gas at temperature T and chemical potential µ. For a finite temperature T and a large volume V , N_th has a smooth behavior as a function of µ and reaches its maximum N_c= N_th(T, µ = ε₀) asymptotically at µ = ε₀ as shown in Fig. 3.1. On the other hand, N₀ diverges when µ is approaching to ε₀. If the value of N_c is lager than N , (3.21) is always satisfied for values of µ considerably smaller than ε₀ and N₀ is negligible compared to N . This corresponds to the point (µ₁, N₁) in Fig. 3.1. If N_cis smaller than N , on the contrary, the occupation number N₀ of the ground state is substantial and thus it is expected the condensate is formed. This corresponds to the point (µ₂, N₂) in Fig. 3.1. Since the ground state population is given by N₀ = £

e^β(ε0−µ)− 1¤₋₁

, the difference between the ground state energy and the chemical potential is substantially smaller than k_BT :

ε₀− µ = k_BT ln µ

1 + 1 N₀

¶

' k_BT

N₀ . (3.23)

0

condensate fraction

50%

occupation number

chemical potential 0

condensate fraction

50%

occupation number

chemical potential

Figure 3.1: The occupation number N₀ in the ground state and N_th in all excited states vs. chemical potential µ. If N > N_c, the system exhibits BEC.

The critical temperature T_c of BEC is operationally defined by the relation

N_th(T_c, µ = ε₀) = N. (3.24) For a system with a finite volume V , the total number of single particle states in the energy range k_BT measured from the ground state energy ε₀is finite, and so N_th(T_c, µ = ε₀) stays a finite value. This means that (3.24) is always satisfied at finite temperatures and the BEC critical temperature exists irrespective of dimensionality. This is in sharp contrast to the opposite conclusions drawn for a uniform 2D or 1D system with infinite area or length [4]. We will discuss this point further in the next section.

3.3 BEC threshold in a uniform system

3.3.1 Energy density of states

The energy density of states for a free particle with a mass m in a d−dimensional system with a system size L is calculated by the standard method

ρ^(d)(ε) = Ω_d µ L

2π

¶_d 1 2

µ2m

~²

¶^d

2 ε^d2−1, (3.25)

where

Ω_d=





4π (d = 3) 2π (d = 2) 1 (d = 1)

(3.26) The normalized energy density of states is plotted for three-, two- and one-dimensional systems in Fig. 3.2. Note that ρ⁽³⁾(ε) vanishes when ε → 0, while ρ⁽²⁾(ε) stays constant and ρ⁽¹⁾(ε) diverges in the same limit. This difference in the asymptotic behaviors of ρ^(d)(ε → 0) results in the important consequence that a finite BEC critical temperature T_c exists only in a three-dimensional system [4]. However, as mentioned already, if a system size is finite, the non-zero BEC critical temperature T_c exists independent of dimensionality.

3.3.2 BEC critical temperature/density

The BEC critical temperature for a uniform 3D system is given by the condition that all particles accommodated in excited single particle states (except for a ground state) when

0 (a)

0 (c)

0 (b)

1

0 (b)

1

Figure 3.2: The normalized energy density of states for uniform 3D, 2D and 1D systems.

µ = ε₀= 0 are equal to the total number of particles in the system:

N = V 4π²

µ2m

~²

¶³

2 Z _∞

0

√ε dε

e^βε− 1. (3.27)

We can evaluate the energy integral using the relation, _ex¹−1 =P_∞

n=1e^−nx, where x = βε:

Z _∞

0

√ε dε

e^βε− 1 = β⁻³² X∞ n=1

Z _∞

0

e^−nxx^1/2dx

= β⁻³² X∞ n=1

n⁻³² Z _∞

0

e^−t√ tdt

= β⁻³²ζ µ3

2

¶ Γ

µ3 2

¶

, (3.28)

where ζ¡₃

2

¢ = P_∞

n=1n⁻³² and Γ¡₃

2

¢ = R_∞

0 e^−t√

tdt. Substituting (3.28) into (3.27), one obtain the BEC critical density for a uniform 3D system,

n_c≡ N_c

V = 1

4π² µ2m

β~²

¶³

2ζ µ3

2

¶ Γ

µ3 2

¶

' 2.612 1

λ³_Tc. (3.29)

Here λ_Tc =

q 2π~²

mkBTc is the thermal de Broglie wavelength at the critical temperature. Since n^−1/3_c is the average distance between particles, the BEC critical condition (3.29) means that BEC occurs when the inter-particle distance becomes comparable to the thermal de Broglie wavelength of the particle at a given temperature. When the temperature is lower than T_c (or equivalently the density is larger than n_c), the mixture of the condensate at ε₀ = 0 (single particle ground state) and the thermal populations at ε > 0 (single particle excited states) is formed:

n = 2.612

λ³_T + n₀. (3.30)

3.3.3 Condensate fraction

The BEC critical condition is often expressed in terms of the Bose function g_p(Z) defined by

g_p(Z) = 1 Γ(p)

Z _∞

0

dxx^p−1 1 Z⁻¹e^x− 1

= X∞

l=1

Z^l

l^p. (3.31)

where Z = e^βµ is a fugacity and Γ(p) = (p − 1)!. The energy integral for uniform 3D, 2D and 1D systems are then reduced to the Bose functions of Z = 1 and p = ³₂, p = 1 and p = 1/2, respectively. Among them, only g³

2(1) converges, while g₁(1) and g_1/2(1) diverge, so that a finite critical temperature T_c6= 0 exists only for a 3D system as far as a system is uniform and infinite [4].

At a critical temperature T_c in a uniform 3D system, all particles are in the thermal populations, i.e. distributed over single particle excited states at ε > 0:

V λ³_Tcg³

2(1) = N, (3.32)

while at lower temperatures than T_c, the thermal population is lower than (3.32) V

λ³_Tg³

2(1) = N_T < N. (3.33)

Taking the ratio of (3.32) to (3.33), one obtains λ³_T λ³_Tc = N

N_T. (3.34)

From this relation, the number of particles in the condensate is expressed as

N₀ = N − N_T = N

"

1 − µT

T_c

¶³

2

#

. (3.35)

Figure 3.3(a) shows the condensate fraction N₀/N vs. normalized temperature T /T_c. If an ideal gas is trapped in a 3D harmonic potential, the above argument must be modified to take into account the new boundary condition. The condensate fraction in this case is not given by (3.35) but given by N₀/N = 1 − (T /T_c)³, which is confirmed by the experimental results shown in Fig. 3.3(b) [5].

3.3.4 Volume requirement for BEC

For a relatively small system, the energy difference between the first excited state and the ground state becomes an appreciable value, ε₁− ε₀ = ~²/2mV^2/3. If the thermal energy k_BT becomes much smaller than ε₁− ε₀, almost all particles occupy the ground state, i.e.

0 1 1

(a) (b)

0 0.6 1.2 1.8

0 0.5

1

Figure 3.3: (a) The condensate fraction N₀/N vs. the normalized temperature T /T_c in a uniform 3D system. (b) The condensate fraction vs. the normalized temperature for an atomic BEC in a 3D harmonic trap [5]. The dashed line is given by N₀/N = 1 − (T /T_c)³.

N_T ¿ N when ~²/2mV^2/3 À k_BT . This should not be considered as BEC. For a system with a large enough volume, the temperature range satisfying

~²/2mV^2/3 ¿ k_BT ¿ k_BT_c, (3.36) can be found, where N_T ¿ N is realized due to the quantum statistical properties of Bose particles. From (3.36) we obtain the volume requirement for BEC as follows:

V À V_c=

µ2mk_BT_c

~²

¶³

2 ∼ λ³_Tc. (3.37)

That is, the system volume must be much larger than the cube of thermal de Broglie wavelength. The use of the inequality ~²/2mV^2/3 ¿ k_BT is actually crucial in the above theory of BEC based on the energy density of states and the continuous energy integral rather than discrete sum over the single particle excited states.

The chemical potential approaches to the ground state energy, µ → ε₀, in the limit of N₀ → ∞ as evidenced by (3.23). Setting ε₀− µ = 0 in the evaluation of N_T is then justified only if ε₁− ε₀ = ~²/2mV^2/3À ε₀− µ ' k_BT /N₀ is satisfied.

3.4 Thermodynamic functions of an ideal gas

The total energy of the system is given by

E = X

i

ε_i e^{[β(εi−µ)]}− 1

= 3

2k_BT V λ³_Tg⁵

2(Z), (3.38)

where g5

2(Z) = ₃^√4_π R_∞

0 dxx^3/2_Z−1¹e^x−1 corresponds to the Bose function (3.31) with p = ⁵₂ and Z = e^βµ. For T < T_c, Z = 1 and one has g⁵

2(1) = 1.342. Then, the specific heat C_V = ^∂E_∂T is obtained as

C_V

k_BN = 15 4

v λ³_Tg⁵

2(1), (3.39)

for T < T_c, and

C_V

k_BN = 15 4

v λ³_Tg⁵

2(Z) − 9 4

g³

2(Z) g1

2(Z), (3.40)

for T > T_c, where v = 1/n = ^V_N is the specific volume. The specific heat has a typical cusp at T_c as shown in Fig. 3.4.

0 1

2

3 1

0 1

2

3 1

Figure 3.4: Specific heat of an ideal uniform Bose gas vs. temperature. For large temper- atures, the curve approaches to the classical value of 3/2.

For ideal gases in three dimensions, the thermodynamic relation P = ²₃^E_V holds [2]. In BEC, the energy E increases linearly with the volume V , so that using (3.38) with Z = 1 for T ≤ T_c, one obtains the equation of state for BEC,

P = k_BT λ³_T g5

2(1). (3.41)

The pressure of the gas does not depend on the volume in a BEC regime so that the compressibility of the BEC phase is infinite. This pathological feature is remedied by the inclusion of the two-body interactions (see Chapter 4). At T > T_c, the pressure is given by

P = k_BT λ³_T g⁵

2(Z). (3.42)

Since Z = e^βµ decreases toward zero with increasing the volume V , the pressure P also decreases with V . Figure 3.5 shows the equation of state of the ideal Bose gas. The phase transition line separating the BEC and normal phases is obtained by substituting (3.32) or equivalently k_BT_c= ^2π~_m²

h vg3

2(1) i_−2/3

into (3.41) as P v^5/3 =¡

2π~²m¢ g⁵

2(1)/g³

2(1)^5/3. (3.43)

3.5 Spatial coherence function

The off-diagonal element of the one-body density operator (2.23), which is a key parameter for understanding the long-range order (or spatial coherence), is given by

g⁽¹⁾(r₁, r₂) = 1 N

X

i

n_iϕ^∗_i (r₁) ϕ_i(r₂) , (3.44)

phase transition line

normal phase

BEC phase

Figure 3.5: Pressure of the ideal Bose gas vs the specific volume v = V /N for two tem- peratures T₁> T₂.

for the ideal Bose gas. Here ϕ_i is the single particle wave function and n_i is the average occupation of the state i and given by (3.14). By separating the contribution of the condensate (i = 0) and that of the thermal component (i 6= 0), (3.44) can be rewritten as

g⁽¹⁾(r₁, r₂) = N₀ N + 1

N V (2π~)³

Z

dp e^{ip·(r1−r2)/~}

e^β(p2/2m)− 1. (3.45) The corresponding momentum distribution is

n(p) = N₀δ(p) + V (2π~)³

1

e^β(p2/2m)− 1. (3.46) The existence of the delta function in the momentum distribution for T ≤ T_c is the clear signature of BEC and is referred to as ”momentum-space condensation”.

It is interesting to see the behavior of the first-order spatial coherence function at large distances s = |r₁− r₂|. This is determined by the small-momentum components of the thermal distribution (long-wavelength fluctuation). For a temperature below the BEC critical temperature T < T_cone can use the low-momentum expansion:

1

exp (βp²/2m) − 1 ∼ 2mk_BT

p² , (3.47)

where p²/2m ¿ k_BT is used. Using (3.47) in (3.45), one obtains the 1/s behavior of g⁽¹⁾(s),

g⁽¹⁾(s) = N₀

N + V

N (2π)³ Z λ²_T ·1

s, (3.48)

where Z = exp(βµ) ' 1 is the fugacity and λ_T is the thermal de Broglie wavelength. The 1/p² dependence in (3.47) and the 1/s dependence of (3.48) are the general consequences of BEC at a finite temperature.

At temperatures close to T_c, one can use the slightly different low-momentum expansion 1

exp [β (p²/2m − µ)] − 1 ∼ Z2mk_BT

p²+ p²_c , (3.49)

where p²_c = 2mk_BT (1 − Z). Using this expression in (3.45) together with N₀ = 0, one obtains

g⁽¹⁾(s) = 1 N

V (2π)³

Z λ²_T

exp h

−p

4π(1 − Z)s/λ_T i

s . (3.50)

The first-order coherence function decays exponentially, where λ_T and Z determine the decay constant.

At temperatures far above T_c, i.e. βµ → −∞, the momentum distribution approaches the classical Maxwell-Boltzmann distribution exp£

−p²/2mk_BT¤

and the first order spatial coherence function(3.45) is reduced to

g⁽¹⁾(s) = exp µ

−πs² λ²_T

¶

. (3.51)

The first-order coherence function decays by a gaussian law. For both cases, (3.50) and (3.51), the first-order spatial coherence function tends to zero within a distance approxi- mately equal to the thermal de Broglie wavelength. However, at T < T_c, the first-order coherence function remains a finite value N₀/N at macroscopic distances due to BEC.

Such behaviors of the cold Bose gas are shown in Fig.2.4.

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Rev., 158:383–386, 1967.

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